What can you say about the function that generated the following table of values?

| [tex]\(x\)[/tex] | [tex]\(y\)[/tex] |
|:-------:|:-------:|
| -4 | -0.906 |
| -2 | -0.625 |
| 0 | 0.5 |
| 2 | 5 |
| 4 | 23 |

A. The function has exactly one [tex]\(x\)[/tex]-intercept.
B. The function has no [tex]\(x\)[/tex]-intercepts.
C. Not enough information to answer the question.
D. The function has more than one [tex]\(x\)[/tex]-intercept.



Answer :

From the given table of values for the function, we can analyze how the function behaves:

[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -4 & -0.906 \\ \hline -2 & -0.625 \\ \hline 0 & 0.5 \\ \hline 2 & 5 \\ \hline 4 & 23 \\ \hline \end{array} \][/tex]

### Step-by-Step Analysis:

1. Examining the y-values for changes in sign:
- When [tex]\( x = -4 \)[/tex], [tex]\( y = -0.906 \)[/tex] (y is negative)
- When [tex]\( x = -2 \)[/tex], [tex]\( y = -0.625 \)[/tex] (y is negative)
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0.5 \)[/tex] (y is positive)

2. Trend of y-values:
- As [tex]\( x \)[/tex] increases from [tex]\(-4\)[/tex] to [tex]\(-2\)[/tex] to [tex]\(0\)[/tex] to [tex]\(2\)[/tex] to [tex]\(4\)[/tex], the y-values are increasing: [tex]\(-0.906, -0.625, 0.5, 5, 23\)[/tex].

3. X-intercept analysis:
- The sign of the y-value changes from negative to positive between [tex]\( x = -2 \)[/tex] and [tex]\( x = 0 \)[/tex]. This indicates that the function crosses the x-axis between these points.
- Given that the sign change happens once and there are no other sign changes observed in the provided data, it suggests the function has exactly one x-intercept between [tex]\( x = -2 \)[/tex] and [tex]\( x = 0 \)[/tex].

Based on these observations, the correct conclusion is:

A. the function has exactly one [tex]\( x \)[/tex]-intercept